[Edu-sig] thought re graphing calculators ...

Gregor Lingl gregor.lingl at aon.at
Wed Sep 30 02:58:47 CEST 2009



kirby urner schrieb:
> On Tue, Sep 29, 2009 at 9:15 AM, Gregor Lingl <gregor.lingl at aon.at> wrote:
>   
>> Strategy of escalation? Arms race?
>>
>>     
>
> Not so much.  There's nothing on the other side.  Will anyone do this
> manually?  Is that what "correctly" means?  More likely they mean
> something like "symbolically" which is akin to "just imagining
> something without really doing any of the work" (so no contest, I walk
> away happy).
>
> The ability to brute force these data points with a self-feedback
> circuit governed by various expressions, is for computers and
> computers only.  Humans by themselves aren't even in the game.  At the
> very least you'll want an abacus, or lowly calculator if you're a nerd
> (snicker).
>   
Oh no, when thinking about calculations or even only viewing diverging 
graphs humans not
only are in the game but are still its main characters.

Since say 5000 years humans have devoloped the concepts of numbers, 
calculations and
algebra. They have discovered, that calculations obey certain algebraic 
laws like
a*(b+c) = a*b + a*c and the like. Finally they have devoloped the 
concepts of
algebraic structures like rings, fields etc.

The purpose of my script simply is to show, that  what we know as real 
numbers are
different things  (entities) than what we have invented (using nature an 
physical phenomena)
as machine numbers. These two simply obey different sets of algebraic 
laws. The distributive
law is not valid for machine numbers with the operations + and *. And 
this statement is true
independent from the setting for getcontext().prec.
(Floating point) machine numbers with + and * do not form a field. To 
describe their
behaviour you have to devise different algebraic structures.

This is not a gewgaw!

In fact my intentions are much less ambituous. I'd be very glad if even 
50 % of my
students accepted seriously that the squareroot of two does not equal to 
1.41421356237.
They do not, regardless of the fact that they are able to multiply this 
number with itself
by hand (!) and to recognize that  the result  does not equal 2.
>> from turtle import *
>> from decimal import Decimal, getcontext
>> getcontext().prec = 50
>>
>> k = Decimal('3.9')
>>
>> N = 250
>>
>> That's what I meant with (in principle)
>>
>> Gregor
>>
>>     
>
> Yes, I understood.  But what's the principle?
>
> In my curriculum, we worship nature and physical phenomena a lot more,
>   
Usually, e. g. when explaining the butterfly effect (does one use this 
term in English?), on argues
that long-term forecasting the future is impossible because of necessary 
inaccuracies of the
initial values as results of measuements. But look precisely: in the 
example we talk about,
there are no measurements, which can be inaccurate and the initial 
values are well defined.
It's the mathematics of machine numbers which generates - after a 
sequence of only a few
hundred arithmetic operations - results, which are completely senseless 
(or lack any meaning).
If you don't mind,  ok. But you have to admit it.

That is not the case with ordinary numbers.
> so this imaginary thing where you imagine the same curves out to
> infinity, but don't actually plot anything, who worthless is that?  
A very dangerous argument, in my opinion. Many (if not most) 
mathematicians agree with
the statement that one of the most central concepts  of mathematics - 
and the one concept
which makes mathematics interesting - is the concept of infinity.
If not one had to abandon  a large part of classical mathematics.
> To think, people actually get paid for such daydreaming.  Amazing.
>   
Now I stared at this sentence for about 10 Minutes but I can't figure 
out what you wanted
to express exactly with this phrase. Don't know if it is my limited 
knowledge of the English language,
my lack of getting the irony or simply my inability to accept the point 
of view of the
modern geometrician, who anly accepts as real what he can visualize.

Gregor
> Kirby
>
>   


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